Problem: All of the 4th grade teachers and students from Covington went on a field trip to an archaeology museum. Tickets were $$7.50$ each for teachers and $$4.00$ each for students, and the group paid $$39.00$ in total. A few weeks later, the same group visited a science museum where the tickets cost $$22.50$ each for teachers and $$8.50$ each for students, and the group paid $$96.00$ in total. Find the number of teachers and students on the field trips.
Explanation: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${7.5x+4y = 39}$ ${22.5x+8.5y = 96}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-3$ ${-22.5x-12y = -117}$ ${22.5x+8.5y = 96}$ Add the top and bottom equations together. $ -3.5y = -21 $ $ y = \dfrac{-21}{-3.5}$ ${y = 6}$ Now that you know ${y = 6}$ , plug it back into $ {7.5x+4y = 39}$ to find $x$ ${7.5x + 4}{(6)}{= 39}$ $7.5x+24 = 39$ $7.5x = 15$ $x = \dfrac{15}{7.5}$ ${x = 2}$ You can also plug ${y = 6}$ into $ {22.5x+8.5y = 96}$ and get the same answer for $x$ ${22.5x + 8.5}{(6)}{= 96}$ ${x = 2}$ There were $2$ teachers and $6$ students on the field trips.